do it in the sql query since you already have the arrival time in the DB
$query = "SELECT PatientID, Forename, Surname, Gender, Illness, Priority, Arrival_Time, TIMEDIFF(Arrival_time,NOW()) as Waiting_Time FROM Patient";
Also in your example above your only ever comparing now time to now time never actually comparing against Arrival_Time from your DB
Example modified version of your code
<?php
$conn = mysqli_connect("localhost","root","") or die ("No connection");
mysqli_select_db($conn, "a&e") or die('Could not select database.');
$query = "SELECT PatientID, Forename, Surname, Gender, Illness, Priority, Arrival_Time , TIMEDIFF(Arrival_time,NOW()) as Waiting_Time FROM Patient";
$result = mysqli_query($conn, $query) or die("Invalid query");
date_default_timezone_set('Europe/London');
echo "<table border='1'>
<tr>
<th>PatientID</th>
<th>Forename</th>
<th>Surname</th>
<th>Gender</th>
<th>Illness</th>
<th>Priority</th>
<th>Waiting Time</th>
</tr>";
while ($row = $result->fetch_object()){
echo "<tr>
<td>" . $row[0] . "</td>
<td>" . $row[1] . "</td>
<td>" . $row[2] . "</td>
<td>" . $row[3] . "</td>
<td>" . $row[4] . "</td>
<td>" . $row[5] . "</td>
<td>" . $row[7] . "</td>
</tr>";
}
echo "</table>";
mysqli_close($conn);
?>